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The Signless Laplacian Estrada Index of Unicyclic Graphs  [PDF]
Hamid Reza Ellahi,Ramin Nasiri,Gholam Hossein Fath-Tabar,Ahmad Gholami
Mathematics , 2014,
Abstract: For a graph $G$, the signless Laplacian Estrada index is defined as $SLEE(G)=\sum^{n}_{i=1}e^{q^{}_i}$,where $q_1, q_2, \dots, q_n$are the eigenvalues of the signless Laplacian matrix of $G$. In this paper, we first characterize the unicyclic graphs with the first two largest and smallest $SLEE$ and then determine the unique unicyclic graph with maximum $SLEE$ among the unicyclic graphs on $n$ vertices with given diameter.
On a conjecture for the signless Laplacian spectral radius of cacti with given matching number  [PDF]
Yun Shen,Lihua You,Minjie Zhang,Shuchao Li
Mathematics , 2015,
Abstract: A connected graph $G$ is a cactus if any two of its cycles have at most one common vertex. Let $\ell_n^m$ be the set of cacti on $n$ vertices with matching number $m.$ S.C. Li and M.J. Zhang determined the unique graph with the maximum signless Laplacian spectral radius among all cacti in $\ell_n^m$ with $n=2m$. In this paper, we characterize the case $n\geq 2m+1$. This confirms the conjecture of Li and Zhang(S.C. Li, M.J. Zhang, On the signless Laplacian index of cacti with a given number of pendant vetices, Linear Algebra Appl. 436, 2012, 4400--4411). Further, we characterize the unique graph with the maximum signless Laplacian spectral radius among all cacti on $n$ vertices.
On the Main Signless Laplacian Eigenvalues of a Graph  [PDF]
Hanyuan Deng,He Huang
Mathematics , 2012,
Abstract: A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, we first give the necessary and sufficient conditions for a graph with one main signless Laplacian eigenvalue or two main signless Laplacian eigenvalues, and then characterize the trees and unicyclic graphs with exactly two main signless Laplacian eigenvalues, respectively.
An upper bound on the largest signless Laplacian of an odd unicyclic graph
Collao,Macarena; Pizarro,Pamela; Rojo,Oscar;
Proyecciones (Antofagasta) , 2012, DOI: 10.4067/S0716-09172012000100005
Abstract: we derive an upper bound on the largest signless laplacian eigenvalue of an odd unicyclic graph. the bound is given in terms of the largest vertex degree and the largest height of the trees obtained removing the edges of the unique cycle in the graph.
An upper bound on the largest signless Laplacian of an odd unicyclic graph  [cached]
Macarena Collao,Pamela Pizarro,Oscar Rojo
Proyecciones (Antofagasta) , 2012,
Abstract: We derive an upper bound on the largest signless Laplacian eigenvalue of an odd unicyclic graph. The bound is given in terms of the largest vertex degree and the largest height of the trees obtained removing the edges of the unique cycle in the graph.
On Maximum Signless Laplacian Estrada Indices of Graphs with Given Parameters  [PDF]
H. R. Ellahi,R. Nasiri,G. H. Fath-Tabar,A. Gholami
Mathematics , 2014,
Abstract: Signless Laplacian Estrada index of a graph $G$, defined as $SLEE(G)=\sum^{n}_{i=1}e^{q_i}$, where $q_1, q_2, \cdots, q_n$ are the eigenvalues of the matrix $\mathbf{Q}(G)=\mathbf{D}(G)+\mathbf{A}(G)$. We determine the unique graphs with maximum signless Laplacian Estrada indices among the set of graphs with given number of cut edges, pendent vertices, (vertex) connectivity and edge connectivity.
Matching Energy of Unicyclic and Bicyclic Graphs with a Given Diameter  [PDF]
Lin Chen,Jinfeng Liu,Yongtang Shi
Mathematics , 2014,
Abstract: Gutman and Wagner proposed the concept of matching energy (ME) and pointed out that the chemical applications of ME go back to the 1970s. Let $G$ be a simple graph of order $n$ and $\mu_1,\mu_2,\ldots,\mu_n$ be the roots of its matching polynomial. The matching energy of $G$ is defined to be the sum of the absolute values of $\mu_{i}\ (i=1,2,\ldots,n)$. In this paper, we characterize the graphs with minimal matching energy among all unicyclic and bicyclic graphs with a given diameter $d$.
On a conjecture for the signless Laplacian eigenvalues  [PDF]
Lihua You,Jieshan Yang
Mathematics , 2013,
Abstract: Let $G$ be a simple graph with $n$ vertices and $e(G)$ edges, and $q_1(G)\geq q_2(G)\geq\cdots\geq q_n(G)\geq0$ be the signless Laplacian eigenvalues of $G.$ Let $S_k^+(G)=\sum_{i=1}^{k}q_i(G),$ where $k=1, 2, \ldots, n.$ F. Ashraf et al. conjectured that $S_k^+(G)\leq e(G)+\binom{k+1}{2}$ for $k=1, 2, \ldots, n.$ In this paper, we give various upper bounds for $S_k^+(G),$ and prove that this conjecture is true for the following cases: connected graph with sufficiently large $k,$ unicyclic graphs and bicyclic graphs for all $k,$ and tricyclic graphs when $k\neq 3.$
On the eccentric distance sum of unicyclic graphs with a given matching number  [PDF]
Shuchao Li,Yibing Song,Bing Wei
Mathematics , 2013,
Abstract: Let $G = (V_G,E_G)$ be a simple connected graph. The eccentric distance sum of $G$ is defined as $\xi^d(G)=\sum_{v \in V_G}\,\varepsilon_G(v)D_G(v),$ where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$ and $D_G(v)=\sum_{u \in V_G}\,d(u,v)$ is the sum of all distances from the vertex $v$. In this paper, we characterize $n$-vertex unicyclic graphs with given matching number having the minimal and second minimal eccentric distance sums, respectively.
On Maximum Signless Laplacian Estrada Index of Graphs with Given Parameters II  [PDF]
Ramin Nasiri,Hamid Reza Ellahi,Gholam Hossein Fath-Tabar,Ahmad Gholami
Mathematics , 2014,
Abstract: Recently Ayyaswamy [1] have introduced a novel concept of the signless Laplacian Estrada index (after here $SLEE$) associated with a graph $G$. After works, we have identified the unique graph with maximum $SLEE$ with a given parameter such as: number of cut vertices, (vertex) connectivity and edge connectivity. In this paper we continue out characterization for two further parameters; diameter and number of cut vertices.
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